We measure the velocity dispersion, {sigma}, and surface density, {Sigma}, of the molecular gas in nearby galaxies from CO spectral line cubes with spatial resolution 45-120pc, matched to the size of individual giant molecular clouds. Combining 11 galaxies from the PHANGS-ALMA survey with four targets from the literature, we characterize ~30000 independent sightlines where CO is detected at good significance. {Sigma} and {sigma} show a strong positive correlation, with the best-fit power-law slope close to the expected value for resolved, self-gravitating clouds. This indicates only a weak variation in the virial parameter {alpha}_vir_{propto}{sigma}^2^/{Sigma}, which is ~1.5-3.0 for most galaxies. We do, however, observe enormous variation in the internal turbulent pressure P_turb_{propto}{Sigma}{sigma}^2^, which spans ~5dex across our sample. We find {Sigma}, {sigma}, and P_turb_ to be systematically larger in more massive galaxies. The same quantities appear enhanced in the central kiloparsec of strongly barred galaxies relative to their disks. Based on sensitive maps of M31 and M33, the slope of the {sigma}-{Sigma} relation flattens at {Sigma}<~10M_{sun}_/pc^2^, leading to high {sigma} for a given {Sigma} and high apparent {alpha}_vir_. This echoes results found in the Milky Way and likely originates from a combination of lower beam-filling factors and a stronger influence of local environment on the dynamical state of molecular gas in the low-density regime.